Numerous applications achieve their results by transforming a given block of data, referred to as a “source block” hereafter. As used here, the word “source block” refers to any data that is stored at one or more sources. Thus, a document, an image, and a file from a file server or computer storage device are all examples of source blocks. Source blocks can be of unknown size (such as a source block taken from the output of a streaming source) or they can be of known size (such as a one megabyte image stored on a hard disk). Either way, a source block is a sequence of one or more source symbols, where each source symbol is a piece of data in the source block that has a position in the source block and a value.
Herein, transformation of a source block refers to an action performed on the source block to achieve a certain result. For example, where a source block is captured by an external device, such as a camera, one possible transformation could amount to the compression of the source block to a size significantly less, to facilitate its storage on smaller storage devices, or to facilitate faster transmission to one or more possible intended recipients. As another example, the source block could be designated for transport over a channel such as a computer network, or a channel in which there is an expectation of corruption or loss. In such a case, the source block may be transformed before transmission so as to increase its robustness with respect to transmission errors.
Of the many applications requiring transformations of source blocks, ones of particular interest are those that perform a transformation to increase the robustness of the source block towards errors incurred in a transmission. Transmission is the process of transmitting a source block from one or more senders to one or more recipients through a channel in order to deliver the source block. If one sender is connected to any number of recipients by a perfect channel, the received data can be an exact copy of the original source block, as all the data will be received correctly. However, where the channel is not perfect, which is the case for most real-world channels, or the data emanates from more than one sender, which is the case for some systems, what is received might not be an exact copy.
Channel imperfections can refer to data erasures, data incompleteness, or data corruption. The act of data transmission does not only refer to the transmission of data between geographically distant locations, and transmission can also include cases in which the data is never physically moved. For example, source blocks stored on a storage medium in which there is a possibility of defects could constitute a form of transmission, as there is a possibility of data corruption when the source block is accessed again.
A common process for protection of source blocks against possible transmission errors is that of coding. With coding, a source block is transformed or a new set of data (sometimes called “redundant” or “repair” data) is computed from the source block. The transformed source block often contains redundant information computed from the original source block, with the goal of using the internal redundancies to obtain information about the errors incurred during the transmission, and correcting such errors. There is a large body of literature on the theory and practice of designing and using codes.
The choice of the code depends on the particular application, and on the communications channel on which transmission is to be conducted. Often, the chosen code has some linearity properties. For example, where the source blocks are made up of one or more source symbols that are groups of bits, the linearity condition would guarantee that the coding of the symbol-wise sum (mapping) of two source blocks is equal to the symbol-wise sum (mapping) of the codings of the source blocks. Such linearity conditions can be used with great advantage to describe and calculate the encoding and the decoding processes. A large sub-class of codes used in practice satisfy such linearity conditions.
The inverse process to the transformation leading to the encoding of a source block is the decoding process. In this process, a (possibly corrupted) version of the encoded source block is processed in such a way as to obtain a good (or sometimes the best possible) estimate of the original state of the source block before transmission.
One of the many benefits of linear coding schemes is the fact that the encoding and decoding processes can be explained by matrices. Matrices are mathematical objects containing entries in form of a two-dimensional array. As is well-known to people of skill in the art, matrices can be conveniently used to represent linear mappings between objects, for example between sets of symbols comprising source blocks.
Often times, the encoding and the decoding process may benefit from the use of additional memory to store intermediate results. For example, some decoding processes may require keeping a copy of the received data in addition to the decoded source block. The amount of additional memory needed for the decoding and encoding processes may be too large on devices with limited memory. For example, where the device is a mobile receiving device, like a mobile phone or a Personal Digital Assistant (PDA), the memory on the device may be small, and/or the memory may have been reserved for other applications intended to run on the device. In such situations, decoding and encoding processes should use memory efficiently, but sometimes this is difficult to implement.